If people are more aware of something, then their awareness of it is greater than it was at some time in the past. If they’re increasingly aware of it, then the rate of growth of their awareness is accelerating. If they are increasingly more aware of it, then the rate of accelertion of their awareness is itself increasing.

I would say that "increasingly more aware" instead implies that the fraction of the people whose awareness is increasing, is also increasing.
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Karl KnechtelJan 28 '12 at 15:34

@Karl Knechtel: I agree - grammatically it's probably nonsense, but when we see "increasingly more" in the context of people's awareness, we tend to assume not only that the number of people who are aware is increasing, but the ones who are already aware are becoming more aware. Or alternatively more people are becoming aware and their levels of awareness are increasing - it doesn't really make much difference since both ways round we're assuming there are two things on the rise at the same time.
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FumbleFingersJan 28 '12 at 16:07